Contains closed set Accumulation points?

[xxxmisko]

Homework Statement

Hello, I am here a novice and my English is very bad. I am a student and now we learning about sets. I have got a problem, how to proof, that every closed set contains all accumulation points? I know / hope, that should, but I want to proof it. I hope, that somebody will help me. Have a nice day, Michal, Slovakia

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

What definition of "closed set" are you using? There are several different but equivalent definitions of "closed set"- and in mathematics, proofs often use the specific words of definitions. No one can help you prove this without knowing what definition you are using.

 

[xxxmisko]

What definition of "closed set" are you using? There are several different but equivalent definitions of "closed set"- and in mathematics, proofs often use the specific words of definitions. No one can help you prove this without knowing what definition you are using.

I do not know, how do you think it, but this is example for my closed set

Closed interval [a,b] is closed subset of real numbers

We have not in school definition of closed and opened set yet. But we are working with functions too, also it should be a definition of intervals of the function -set of function of definite domain? I really do not know. I hope, that do you understand me. Thanks for your fast answer

 

[HallsofIvy]

If you do not know the definition of "closed set", then you cannot possibly prove anything about them! Examples will not provide proofs. Since you have a specific definition of "closed interval" you might be able to prove that a closed interval contains all its accumulation points. What is your definition of "accumulation point"?

 

[xxxmisko]

If you do not know the definition of "closed set", then you cannot possibly prove anything about them! Examples will not provide proofs. Since you have a specific definition of "closed interval" you might be able to prove that a closed interval contains all its accumulation points. What is your definition of "accumulation point"?

Let (zero) ∅≠M⊂R. Point a∈R is accumulation point of set M, if for every O_{ε}°(a) exists x∈M, x∈O_{ε}°(a).

I hope that there will be no problem with syntax, because I am not using LaTex yet